D. Blackwell

English

Research Papers in Economics

AN ANALOG OF THE MINIMAX THEOREM FOR
VECTOR PAYOFFS
DAVID BLACKWELL
1. Introduction* The von Neumann minimax theorem [2] for finite
games asserts that for every rxs matrix M=\\m(i, j)\\ with real elements
there exist a number v and vectors
P=(Pi, •••, Pr)f Q={QU •••> Qs)f Pi, Qj>β,
such that
i> 3)
for all i, j. Thus in the (two-person, zero-sum) game with matrix Λf,
player I has a strategy insuring an expected gain of at least v, and
player II has a strategy insuring an expected loss of at most v. An
alternative statement, which follows from the von Neumann theorem
and an appropriate law of large numbers is that, for any ε>0, I can,
in a long series of plays of the game with matrix M, guarantee, with
probability approaching 1 as the number of plays becomes infinite, that
his average actual gain per play exceeds v — ε and that II can similarly
restrict his average actual loss to v-he. These facts are assertions about
the extent to which each player can control the center of gravity of
the actual payoffs in a long series of plays. In this paper we investigate
the extent to which this center of gravity can be controlled by the
players for the case of matrices M whose elements m(i9 j) are points
of ΛΓ-space. Roughly, we seek to answer the following question. Given
a matrix M and a set S in iV-space, can I guarantee that the center of
gravity of the payoffs in a long series of plays is in or arbitrarily near
St with probability approaching 1 as the number of plays becomes in-
finite ? The question is formulated more precisely below, and a complete
solution is given in two cases: the case JV=1 and the case of convex S.
Let
Λf = \\m(i, j) 1, l^i^r, l^j^s
be an rxs matrix, each element of which is a probability distribution
over a closed bounded convex set X in Euclidean iV-space. By a
strategy for Player I is meant a sequence /={/„}, n=0, 1, 2, ••• of
functions, where fn is defined on the set of ^-tuples (xu •••, xn), xteX
Received September 14, 1954. This paper was written under contract Nonr 1197(00)
with the Office of Naval Research.2 DAVID BLACKWELL
and has values in the set P of vectors p=(Pu , pr) with
1 /o is simply a point in P. A strategy g= {gn} for Player II is defined
similarly, except that the values of gn are in the set Q of vectors q=
(Qu •—, Qs) with qj^O, Σf^=l. The interpretation is that I, II select
i, j according to the distributions /0, g0 respectively, and a point x±eX is selected according to the distribution m(i, j). The players are told
xl9 after which they again select i, j, this time according to the distri-
butions fi(xi), gι{%ι), a point x2 is chosen according to the m(i, j) cor-
responding to their second choices, they are told x2 and select a third
i, j according to f%(xly #2), g*{xu x2), etc. Thus each pair (/, g) of strategies, together with M, determines a sequence of (vector-valued)
random variables xu x2, .
Let S be any set in iV-space. We shall say that S is approachable
with /* in M, if for every e>0 there is an JV0 such that, for every g,
Prob {dn^>ε for some n^>NQ} <ε ,
where δn denotes the distance of the point Σϊ
χil
n from S and xu x2,
• are the variables determined by /*, g. We shall say that S is ex-
cludable with g* in M, if there exists d>0 such that for every ε>0
there is an NQ such that, for every /,
Prob {δ^d for all rc^
where xlf x%, ••• are the variables determined by /, g*. We shall say
that & is approachable (excludable) in M, if there exists /* (g*) such
that S is approachable with /* (excludable with g*). Approachability
and excludability are clearly the same for S and its closure, so that we
may suppose S closed.
In terms of these concepts, von Neumann's theorem has the follow-
ing analog.
For N=l, associated with every M are a number v and vectors pe P,
qeQ such that the set S= {#£>£} is approachable for t<Lv with f:fn=p
and excludable for ty>v with g : gn=q.
A slightly more complete result for N=l, characterizing all ap-
proachable and excludable sets S for a given M, is given in § 4 below.
Obviously any superset of an approachable set is approachable, any
subset of an excludable set is excludable, and no set is both approach-
able and excludable. Another obvious fact which will be useful is that
if a closed set & is approachable in the sxr matrix Λf', the transpose
of jfcf, then any closed set T not intersecting S is excludable in M with
any strategy with which S is approachable in M'. Thus any sufficient
condition for approachability yields immediately a sufficient condition for
excludability. A sufficient condition for approachability is given in § 2.
It turns out that every convex S satisfies either this condition forAN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS 3
approachability or the corresponding condition for excludability, enabling
us to give in § 3 a complete solution for convex S. For non-convex S,
the problem is not solved except for 2V=1. An example of a set which
is neither approachable nor excludable in a given M is given in § 5, the
concepts of weak approachability and excludability are introduced, and
it is conjectured that every set is either weakly approachable or weakly
excludable.
2 A sufficient condition for approachability. If x9 y are distinct
points in iV-space, H is the hyperplane through y perpendicular to the
line segment xy, and z is any point on H or on the opposite side of H
from xy then all points interior to the line segment xz and sufficiently
near x are closer to y than is x. This fact is the basis for our sufficient
condition for approachability.
For any matrix M, denote by M the matrix whose elements m{i, j)
are the mean values of the distributions m(i, j). For any peP denote
by R(p) the convex hull of the s points Σ« Pιm(i, j). The sufficient
condition for approachability is given in the following theorem.
THEOREM 1. Let S be any closed set. If for every xφS there is a
p (=p(x))e P such that the hyperplane through y, the closest point in S
to x, perpendicular to the line segment xy separates x from R{p), then
S is approachable with the strategy f:fn, where
fn-
xn) if n>0 and xn = (^ \n
^arbitrary if n=0 or xneS.
Proof. Suppose the hypotheses satisfied, let I use the specified
strategy, let II use any strategy, and let xlf x2, be the resulting
sequence of chance variables. For
let yn be the point of S closest to xn, and write un=yn — xn. Then, for
(1) E((un, xn+
where E(x\y) denotes the conditional expectation of x given y and (u, v)
denotes the inner product of the vectors u and v.
Let δn denote the squared distance from xn to S. If <5w>0, then
(2) dn+1<:\xn+1-yn\
2^\xn-yn\
2 + 2(xn-ynf xn+1-xn)+ \xn+ι-xn\\4 DAVID BLACKWELL
Since xn+i—Sn=(a?n+i —5n)/(w + l), we have
/ Q \ (7γ. n, ™ ™ \ fan Vn> •^n + l^^Vn) i fan Vn* Vn ^n)
71 + 1 92 + 1
and
(4) l»«+i-Sn
where c depends only on the size of the bounded set X. From (2), using
(1), (3), and (4), we obtain, replacing n by n—1,
(5) E{dn\δl9 -.., an_1)(5 \ n / n
2
Moreover
(6) O^r^α
and
(7) |3n-3»-il^-.
7Z
Thus it remains only to establish the following.
LEMMA. A sequence of chance variables δlf δ2, satisfying (5),
(6), and (7) converges to zero with probability 1 at a rate depending only
on a, δ, c, that is, for every ε>0 there is an NQ depending only on ε,
α, bf c such that for auy {Sn} satisfying (5), (6), and (7), we have
Prob {δn^>e for some
Proof of Lemma. Let rc0 be any integer. There exists
depending only on nQf ε, a, c such that
Prob {<5w:>ε/2 for n^rι<snύ<e\
<l.
To see this, define, for n^>n0, an=Sn if <54>O for no<Li<ln, and ^w=0 otherwise. Then an<Cej2 implies <5*<ε/2 for some i with no<Li<Ln. Also
and, for
so that
E(an\ano, , w)^(:,
0 V n J n
ι
nAN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS 5
Thus E(ocn)->0 at a rate depending only on n0, a, c, and there is an nx depending only on n0, ε, α, c for which E(<xni) is so small that
Prob {αWl<ε/2}>l-(ε/2).
For every n, k with n<Lk we define variables znk as follows. Un-
less ίn_i<e/2 and dn>ej2, znk=0 for all k. If δn^<εj2 and <?^ε/2 for n<J,<Jc, then znk=δk. If <ViO/2, 3*^e/2 for n<ji<kQ and ίfco<e/2, then zwfc4-zwfco=<5fcO
 fo
r &I>
feo. If δw^ε for some w^?ii, either <5wI>ε/2
for all w such that no<Ln<L?ι1 or znfci>ε for some ΎQ^UQ. The former event has already been shown to have probability less than ε/2 it
remains to show that the probability of the latter event can be made
less than 6/2 by choosing n0 sufficiently large.
Fix f£>n0 and write βjc=znk—zn fc_χ, &>rc, £n=0. Then, if zn fc-i^ε/2
for sufficiently large ?z0 depending on c and ε, and IβJ <*&/&. If «»»-i<C
ε/2, /5fc=0 so that, in any case
(8) E{βΛβn,~-,βκ-ι)^-fhmz*(\βΛβn, •••,&-!).
We now apply the following form of the strong law of large
numbers, recently proved by the writer [1].
THEOREM 2. If zlf z2, is a sequence cf random variables such
that |3*|<y. and
E(zk\zlf ••••, ^.O^-
then for all t,
Prob K+ +**;>£ for some fc}^(—--Y
1
The variables zk=(nlb)βk^n+1 satisfy the hypotheses of Theorem 2, with w=(ε/2δ), so that
/I \l/6
Prob {«„*-«„„>* for some k>
For large w0, ^ww<3ε/4, so that ^TOΛ^ε for some k implies s«*—sflfι>e/4. Thus
Prob fefc^ε for some
where s=r
ε/4, so thatDAVID BLACKWELL
Prob {zn3bi>e for some ri^n0, ΛC>w}<^
which will be less than ε/2 for n0 sufficiently large. This completes the
proof.
3 The case of convex S.
THEOREM 3. Let T{q) denote the convex hull of the r points
Σί Qjϊniif i) A closed convex set S is approachable if and only if it
intersects every set T(q). If it fails to intersect T(q0), it is excludable
with g: gn^q0.
Proof Suppose S intersects every T(q), let xQ $ S, let y be the point
of S closest to x09 and consider the game with matrix A=\\a(i, j)\\, where
α(i, i)=(2/—a?o> ϊn(i,j)) Its value is
min max {y — xQy Σ Qjmfi* j))=mm max (y—x0, t)^>min (y—xo,s). q i j q teTQq) sGS
Consequently there is a p e P such that
(V-Xof Σ Pim(i, i))^min (y-xOf s)
ί ses
for all j, that is,
(y-x0, r)^>(y-x0, y)
for all reR{p). Since (y—xQ, Xo)<C(y—Xo, y), the hyperplane (y—x0, x)=
(y—Xo, y) separates x0 from R(p), completing the proof.
On the other hand, any T(q0) satisfies the hypotheses of Theorem 1
in M' with f:fn=q09 and so is approachable in M' with this /. Con-
sequently, if S fails to intersect T(qQ), S is excludable in M with
9 -
COROLLARY 1. The sets Rip) are approachable with f:fn~p.
COROLLARY 2. A closed convex set S is approachable if and only if
for every vector u,
)^>min (u, s),
where viu) is the value of the game with matrix |(%, mii, j))\.
Proof of Corollary 2. If for some u0 the inequality fails, then T(q0)
is disjoint from S, where q0 is a good strategy for II in the game with
matrix \\(uOf m(i9 j))\\f and conversely if any T(q0) is disjoint from Sand uQ is a vector withAN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS
max (uQ, £)<Cmin (uQ, s),
then
v(uQ)< min (u0, s). seS
4. The case JV=
THEOREM 4. .For iV=l, te£ v, t;' δe the values of the games with
matrices M, M
f. If v
f<Lv, a closed set S is approachable if it inter-
sects the closed interval v
fv and excludable otherwise. If v'^v, a closed
set S is approachable if it contains the closed interval vv' and excludable
otherwise.
Proof. Application of Corollary 2 to the closed interval AB, A<Cβ
with %=±1 yields that AB is approachable if and only if v^>A and
— v'^>—B. If v'<Lv, these are simply the conditions that AB intersect
the closed interval v'v, and if v'^>v, they are the conditions that AB
contain vv
f. Thus if v'<^v every point in v'v is approachable, so that
any set S intersecting v'v contains an approachable subset and is hence
approachable, while if v'^>v, the interval vv' and hence any set con-
taining it, is approachable. The last sentence, applied to M', yields
that if v'<Lv9 the interval v'v is approachable in M\ so that any closed
set not intersecting v'v is excludable in M, and that if v'^>v, any point
in vv' is approachable in M' so that any closed set not containing vv'
is disjoint from a point approachable in M' and consequently is exclud-
able in M. This completes the proof.
5 An example. We saw in the last section that for N=l every
set is approachable or excludable. This is false for iV=2 as is shown
by the following example. Let
r==s=2, m(l, l)=m(l, 2)=(0, 0), m(2, 1)=(1, 0), m(2, 2)=(1, 1),
let I1 be the set of points (J, y), 0<^/<I~ϊ, let 72 be the set of points
(1> y)> τ^V^Xy and let S=Iι\JI2. For every n, player I has a strategy
which guarantees that x2n e S, as follows: Λ^(0, 1) for j<jn9 so that
xn=(u, 1); if u^i, Λ=(0, 1) for £>rc, and if u<h fj=(h 0) for j>n.
Then for &i>J, x2n e 72, and for u<Ci, x2n e Ix. However S is not approach- able, since the following strategy for II does permit xn to remain near
either I1 or J2. Let xn={any δj, if oQ>§, gn=(l, 0) if an<&, flrn=(0, 1). Thus S is neither approachable nor excludable.
In the above example, S is weakly approachable, where a set £ is
said to be weakly approachable in M if for every ε^>0 there is an No
such that for every n^N0 there is a strategy / for I such that, for
all g,8 DAVID BLACKWELL
ProbR>ε}<ε,
where δn is the distance from xn to S. Similarly S is weakly excludable
in M if there is a cf>0 such that for every e>0 there is an No such
that for every n^>_N0 there is a strategy g for II such that, for all /,
Prob{ίn<d}<e.
Clearly no S is both weakly approachable and weakly excludable , we
conjecture that every S is one or the other. In the above example, it
is not hard to show that a closed S is weakly approachable if it inter-
sects the graph of every function h defined for 0<l£<;i which satisfies
and is weakly excludable if there is such an h whose graph it fails to
intersect.
REFERENCES
1. David Blackwell, On optimal systems, Ann. Math. Stat., 25 (1954), 394-397.
2. John von Neumann and Osker Morgenstern, Theory of games and economic behavior,
Princeton, 1944.
HOWARD UNIVERSITY

On optimal systems,

Theory of games and economic behavior,

An Analog of the Minmax Theorem for Vector Payoffs

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An Analog of the Minmax Theorem for Vector Payoffs

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